Thorup-Zwick Emulators are Universally Optimal Hopsets

A $(β,ε)$-$\textit{hopset}$ is, informally, a weighted edge set that, when added to a graph, allows one to get from point $a$ to point $b$ using a path with at most $β$ edges ("hops") and length $(1+ε)\mathrm{dist}(a,b)$. In this paper we observe that Thorup and Zwick's $\textit{sublinear additive}$ emulators are also actually $(O(k/ε)^k,ε)$-hopsets for every $ε>0$, and that with a small change to the Thorup-Zwick construction, the size of the hopset can be made $O(n^{1+\frac{1}{2^{k+1}-1}})$. As corollaries, we also shave "$k$" factors off the size of Thorup and Zwick's sublinear additive emulators and the sparsest known $(1+ε,O(k/ε)^{k-1})$-spanners, due to Abboud, Bodwin, and Pettie.